May 8 - Lec 1

week Breakdown weeks i y univariatet generating functions s s s multivariate conditional 9 in a stochasticprocess Lawsof probability Random Expirement as long as outcomes are uncertain samplespace n setofall possible outcomes of a random expirement Ex Toss a coin twice a Iii ItIt all outcomes are equally limey it É number of heads outcome arenot equally incley Event subspace ofthe sample space a atleast one Heads HitHeHt Probability is a function that maps events to thereallife satisfying the properties Axiom as pinso proto Axiom a ItAnAa an are mut any exclusive then Pc ai Pai Pebble Intepretation Tmspace Each pebble hasa weight wi w EE wit wa classical Detination pembles in If the weight are equal a w I Yin of Pebble w s y Ex total of Laws of Probability is A a pass i peas A one s fiftiespitta peas peas pears i peas 2 A EB Pease RB B Aoe B A passpass pay non negative B pass apea

Howto Add up two events Peau B Disjoint Peas teas otherwise Peau B pea PCBS PLAN B Hwa on Wapo PCA RB PlanB mut any exclusive Peau Bud past PCBs Pee Plan B O T pianistpeanestpanes Plan Bnc Example Da Montfort'sproblem 4 letters 4 envelopes letters are put in envelopes randomly plat leastone letter is in theright envelope let Ai event where theai letter goes inthe in envelope want peau as u as u au pasta cant pass Peau peanaan trainas painaust pasta nasi put anaust putsnays pea naanas peanaanaustrain assault Plan III Patina as ay 2043044 414 1211 1 For n envelopes I Taylors polyana paina onas e it I t e t t I 5 then as n so prob si te How to multiply events conditional probability we have a probabiities A and B B occurred thennow wouldwe update our probability ofA patio prob ofa given Boccured put unconditional probability pcaiBl pq I_adjuststafnpies.p ace past PEEB gyp

Example Two cards aredrawn without replacement A 1st cardis a Heart B and cardis red findPat 1B and pain B'IFB puts then pants gg P'first Heart whatwe know PCBs Information matters not I time gypy Notes 1 peal B Pobla 21 Information matters not time Lawoftotal Probability cop Ba Ba Be be a partition ofthe sample space planB is Bag Let it beany event PCA PCA IBD PCB Peal Ba PlBa Iz praise PcBr Example Twocoins tanpins m onecoin's chosen at random and tossed sames wantplan three are heads A getting steads Bi first coinwas chosen Bo second coinwas chosen Partitions Bit Bo Plas peal Bi plant pea Ba Ba 123 1413 Example money Hall